math.square on complex, real part

Percentage Accurate: 94.1% → 98.6%

Time: 1.9s

Alternatives: 4

Speedup: 0.6×

• 43.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ re \cdot re - im \cdot im \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ re \cdot re - im \cdot im \end{array} \]

FPCore

(FPCore re_sqr (re im) :precision binary64 (- (* re re) (* im im)))

C

double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Fortran

real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = (re * re) - (im * im)
end function

Java

public static double re_sqr(double re, double im) {
	return (re * re) - (im * im);
}

Python

def re_sqr(re, im):
	return (re * re) - (im * im)

Julia

function re_sqr(re, im)
	return Float64(Float64(re * re) - Float64(im * im))
end

MATLAB

function tmp = re_sqr(re, im)
	tmp = (re * re) - (im * im);
end

Wolfram

re$95$sqr[re_, im_] := N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= re 5e+253) (fma re re (* im (- im))) (* re re)))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if (re <= 5e+253) {
		tmp = fma(re, re, (im * -im));
	} else {
		tmp = re * re;
	}
	return tmp;
}

Julia

re = abs(re)
function re_sqr(re, im)
	tmp = 0.0
	if (re <= 5e+253)
		tmp = fma(re, re, Float64(im * Float64(-im)));
	else
		tmp = Float64(re * re);
	end
	return tmp
end

Wolfram

NOTE: re should be positive before calling this function
re$95$sqr[re_, im_] := If[LessEqual[re, 5e+253], N[(re * re + N[(im * (-im)), $MachinePrecision]), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.9999999999999997e253

   1. Initial program 95.9%

      \[re \cdot re - im \cdot im \]
   2. Step-by-step derivation

      1. sqr-neg 95.9%

         \[\leadsto re \cdot re - \color{blue}{\left(-im\right) \cdot \left(-im\right)} \]

      2. cancel-sign-sub 95.9%

         \[\leadsto \color{blue}{re \cdot re + im \cdot \left(-im\right)} \]

      3. fma-def 99.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)} \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)} \]

   if 4.9999999999999997e253 < re

   1. Initial program 72.7%

      \[re \cdot re - im \cdot im \]

   2. Taylor expanded in re around inf 100.0%

      \[\leadsto \color{blue}{{re}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \color{blue}{re \cdot re} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{re \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(re, re, im \cdot \left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 2: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;re \cdot re \leq 5.4 \cdot 10^{-138} \lor \neg \left(re \cdot re \leq 108000000\right) \land re \cdot re \leq 2.4 \cdot 10^{+117}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (or (<= (* re re) 5.4e-138)
         (and (not (<= (* re re) 108000000.0)) (<= (* re re) 2.4e+117)))
   (* im (- im))
   (* re re)))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if (((re * re) <= 5.4e-138) || (!((re * re) <= 108000000.0) && ((re * re) <= 2.4e+117))) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Fortran

NOTE: re should be positive before calling this function
real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((re * re) <= 5.4d-138) .or. (.not. ((re * re) <= 108000000.0d0)) .and. ((re * re) <= 2.4d+117)) then
        tmp = im * -im
    else
        tmp = re * re
    end if
    re_sqr = tmp
end function

Java

re = Math.abs(re);
public static double re_sqr(double re, double im) {
	double tmp;
	if (((re * re) <= 5.4e-138) || (!((re * re) <= 108000000.0) && ((re * re) <= 2.4e+117))) {
		tmp = im * -im;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

re = abs(re)
def re_sqr(re, im):
	tmp = 0
	if ((re * re) <= 5.4e-138) or (not ((re * re) <= 108000000.0) and ((re * re) <= 2.4e+117)):
		tmp = im * -im
	else:
		tmp = re * re
	return tmp

Julia

re = abs(re)
function re_sqr(re, im)
	tmp = 0.0
	if ((Float64(re * re) <= 5.4e-138) || (!(Float64(re * re) <= 108000000.0) && (Float64(re * re) <= 2.4e+117)))
		tmp = Float64(im * Float64(-im));
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

re = abs(re)
function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if (((re * re) <= 5.4e-138) || (~(((re * re) <= 108000000.0)) && ((re * re) <= 2.4e+117)))
		tmp = im * -im;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: re should be positive before calling this function
re$95$sqr[re_, im_] := If[Or[LessEqual[N[(re * re), $MachinePrecision], 5.4e-138], And[N[Not[LessEqual[N[(re * re), $MachinePrecision], 108000000.0]], $MachinePrecision], LessEqual[N[(re * re), $MachinePrecision], 2.4e+117]]], N[(im * (-im)), $MachinePrecision], N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 re re) < 5.40000000000000057e-138 or 1.08e8 < (*.f64 re re) < 2.3999999999999999e117

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]

   2. Taylor expanded in re around 0 87.8%

      \[\leadsto \color{blue}{-1 \cdot {im}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 87.8%

         \[\leadsto -1 \cdot \color{blue}{\left(im \cdot im\right)} \]

      2. mul-1-neg 87.8%

         \[\leadsto \color{blue}{-im \cdot im} \]

      3. distribute-rgt-neg-in 87.8%

         \[\leadsto \color{blue}{im \cdot \left(-im\right)} \]

   4. Simplified 87.8%

      \[\leadsto \color{blue}{im \cdot \left(-im\right)} \]

   if 5.40000000000000057e-138 < (*.f64 re re) < 1.08e8 or 2.3999999999999999e117 < (*.f64 re re)

   1. Initial program 90.6%

      \[re \cdot re - im \cdot im \]

   2. Taylor expanded in re around inf 78.2%

      \[\leadsto \color{blue}{{re}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 78.2%

         \[\leadsto \color{blue}{re \cdot re} \]

   4. Simplified 78.2%

      \[\leadsto \color{blue}{re \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \cdot re \leq 5.4 \cdot 10^{-138} \lor \neg \left(re \cdot re \leq 108000000\right) \land re \cdot re \leq 2.4 \cdot 10^{+117}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 3: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 5 \cdot 10^{+298}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im)
 :precision binary64
 (if (<= (* im im) 5e+298) (- (* re re) (* im im)) (* im (- im))))

C

re = abs(re);
double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 5e+298) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Fortran

NOTE: re should be positive before calling this function
real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im * im) <= 5d+298) then
        tmp = (re * re) - (im * im)
    else
        tmp = im * -im
    end if
    re_sqr = tmp
end function

Java

re = Math.abs(re);
public static double re_sqr(double re, double im) {
	double tmp;
	if ((im * im) <= 5e+298) {
		tmp = (re * re) - (im * im);
	} else {
		tmp = im * -im;
	}
	return tmp;
}

Python

re = abs(re)
def re_sqr(re, im):
	tmp = 0
	if (im * im) <= 5e+298:
		tmp = (re * re) - (im * im)
	else:
		tmp = im * -im
	return tmp

Julia

re = abs(re)
function re_sqr(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 5e+298)
		tmp = Float64(Float64(re * re) - Float64(im * im));
	else
		tmp = Float64(im * Float64(-im));
	end
	return tmp
end

MATLAB

re = abs(re)
function tmp_2 = re_sqr(re, im)
	tmp = 0.0;
	if ((im * im) <= 5e+298)
		tmp = (re * re) - (im * im);
	else
		tmp = im * -im;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: re should be positive before calling this function
re$95$sqr[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 5e+298], N[(N[(re * re), $MachinePrecision] - N[(im * im), $MachinePrecision]), $MachinePrecision], N[(im * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 5.0000000000000003e298

   1. Initial program 100.0%

      \[re \cdot re - im \cdot im \]

   if 5.0000000000000003e298 < (*.f64 im im)

   1. Initial program 77.2%

      \[re \cdot re - im \cdot im \]

   2. Taylor expanded in re around 0 91.2%

      \[\leadsto \color{blue}{-1 \cdot {im}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 91.2%

         \[\leadsto -1 \cdot \color{blue}{\left(im \cdot im\right)} \]

      2. mul-1-neg 91.2%

         \[\leadsto \color{blue}{-im \cdot im} \]

      3. distribute-rgt-neg-in 91.2%

         \[\leadsto \color{blue}{im \cdot \left(-im\right)} \]

   4. Simplified 91.2%

      \[\leadsto \color{blue}{im \cdot \left(-im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \leq 5 \cdot 10^{+298}:\\ \;\;\;\;re \cdot re - im \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-im\right)\\ \end{array} \]

Alternative 4: 53.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} re = |re|\\ \\ re \cdot re \end{array} \]

FPCore

NOTE: re should be positive before calling this function
(FPCore re_sqr (re im) :precision binary64 (* re re))

C

re = abs(re);
double re_sqr(double re, double im) {
	return re * re;
}

Fortran

NOTE: re should be positive before calling this function
real(8) function re_sqr(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    re_sqr = re * re
end function

Java

re = Math.abs(re);
public static double re_sqr(double re, double im) {
	return re * re;
}

Python

re = abs(re)
def re_sqr(re, im):
	return re * re

Julia

re = abs(re)
function re_sqr(re, im)
	return Float64(re * re)
end

MATLAB

re = abs(re)
function tmp = re_sqr(re, im)
	tmp = re * re;
end

Wolfram

NOTE: re should be positive before calling this function
re$95$sqr[re_, im_] := N[(re * re), $MachinePrecision]

Derivation

1. Initial program 94.9%

   \[re \cdot re - im \cdot im \]

2. Taylor expanded in re around inf 54.7%

   \[\leadsto \color{blue}{{re}^{2}} \]
3. Step-by-step derivation

   1. unpow2 54.7%

      \[\leadsto \color{blue}{re \cdot re} \]

4. Simplified 54.7%

   \[\leadsto \color{blue}{re \cdot re} \]

5. Final simplification 54.7%

   \[\leadsto re \cdot re \]

Reproduce

herbie shell --seed 2023271 
(FPCore re_sqr (re im)
  :name "math.square on complex, real part"
  :precision binary64
  (- (* re re) (* im im)))